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Working together, Katherine and Marina can sand a large cabinet in 2 hours. It would take Katherine 10 hours to do the job alone.

A table showing Rate in part per hour, Time in hours, and Part of Cabinet Sanded. The first row shows, Katherine and has, StartFraction 1 Over 10 EndFraction, 2, and StartFraction 2 Over 10 EndFraction. The second row shows, Marina and has, r, 2, and 2 r.
What is the value of r, the part of the job that Marina can complete in 1 hour?

A. 0.1
B. 0.4
C. 0.5
D. 0.6

1 answer

To solve for the rate \( r \) (the part of the job that Marina can complete in 1 hour), we can start by using the information given in the problem.

  1. First, establish Katherine's rate:

    • Katherine can complete the job alone in 10 hours, so her rate is: \[ \text{Rate of Katherine} = \frac{1}{10} \text{ of the job per hour} \]

  2. Now, look at the combined rate of Katherine and Marina:

    • Together, they can complete the job in 2 hours, so their combined rate is: \[ \text{Combined Rate} = \frac{1}{2} \text{ of the job per hour} \]

  3. Let \( r \) be the rate at which Marina can complete the job:

    • Therefore, Marina's rate can be expressed as: \[ \text{Rate of Marina} = r \text{ of the job per hour} \]

  4. Since their combined rate is equal to the sum of their individual rates, we can set up the equation: \[ \text{Rate of Katherine} + \text{Rate of Marina} = \text{Combined Rate} \] Substituting the known rates into the equation: \[ \frac{1}{10} + r = \frac{1}{2} \]

  5. To solve for \( r \), first clear the fractions by finding a common denominator, which is 10: \[ \frac{1}{10} + r = \frac{5}{10} \] Rearrange the equation: \[ r = \frac{5}{10} - \frac{1}{10} = \frac{4}{10} = 0.4 \]

Thus, the value of \( r \) is \( 0.4 \), meaning Marina can complete \( 0.4 \) of the job in 1 hour.

The answer is \( \boxed{0.4} \).